The Brezis–Nirenberg type problem for the p-laplacian (1 < p < 2): Multiple positive solutions

Abstract

In this paper we consider the quasilinear critical problem(Pλ){−Δpu=λuq−1+up⁎−1inΩu>0inΩu=0on∂Ω where Ω is a regular bounded domain in RN, N≥p2, 1<p<2, p≤q<p⁎, p⁎=Np/(N−p), λ>0 is a parameter. In spite of the lack of C2 regularity of the energy functional associated to (Pλ), we employ new Morse techniques to derive a multiplicity result of solutions. We show that there exists λ⁎>0 such that, for each λ∈(0,λ⁎), either (Pλ) has P1(Ω) distinct solutions or there exists a sequence of quasilinear problems approximating (Pλ), each of them having at least P1(Ω) distinct solutions. These results complete those obtained in [23] for the case p≥2.